Geometric invariant theory for graded additive groups

Abstract

We consider geometric invariant theory for \emph{graded additive groups}, groups of the form ${\mathbb{G}}_a^r{⋊}_w{\mathbb{G}}_m$ such that the ${\mathbb{G}}_m$-action on ${\mathbb{G}}_a^r$ is a scalar multiplication with weight $w\in {\mathbb{N}}_{+}$. We provide an algorithm of equivariant birational modifications, such that we can apply the geometric invariant theory of Bérczi-Doran-Hawes-Kirwan. In particular, the geometric ${\mathbb{G}}_a^r$-quotient exists. This complements Bérczi-Doran-Hawes-Kirwan, in the special case of one grading weight.

Journal of Algebra